Detection of Topological Spin Textures via Nonlinear Magnetic Responses

Topologically nontrivial spin textures, such as skyrmions and dislocations, display emergent electrodynamics and can be moved by spin currents over macroscopic distances. These unique properties and their nanoscale size make them excellent candidates for the development of next-generation race-track memory and unconventional computing. A major challenge for these applications and the investigation of nanoscale magnetic structures in general is the realization of suitable detection schemes. We study magnetic disclinations, dislocations, and domain walls in FeGe and reveal pronounced responses that distinguish them from the helimagnetic background. A combination of magnetic force microscopy (MFM) and micromagnetic simulations links the response to the local magnetic susceptibility, that is, characteristic changes in the spin texture driven by the MFM tip. On the basis of the findings, which we explain using nonlinear response theory, we propose a read-out scheme using superconducting microcoils, presenting an innovative approach for detecting topological spin textures and domain walls in device-relevant geometries.

][11][12] Skyrmions are intriguing as they are nanoscale objects that efficiently couple to spin currents, enabling high storage density and low-energy control. 3,5,13With the progress of the field, the scope widened and other spin textures, such as merons, 14 biskyrmions 15 and hopfions 16,17 have been considered.9][20] The emergence of these topological spin textures is enabled by the lamellar-like morphology of the helimagnetic order analogous to, e.g., cholesteric liquid crystals, 21 swimming bacteria, 22 and the skin on our palms. 23In magnetism, certain analogies exist to ferromagnetic stripe domains 24 , but the involved length scales are substantially different.In chiral magnets, the spin structure twists continuously and the periodicity is up to three orders of magnitude smaller than for the classical stripe domains. 18Edge dislocations within the helimagnetic structure are formed by a pair of + and − disclinations anddepending on their Burgers vector -can carry a topological charge  = − .Such dislocations are topologically equivalent to half-skyrmions or merons as discussed in ref. 19 .Both disclinations and edge dislocations arise even without the external magnetic field usually needed to stabilize skyrmions and represent important building blocks for the formation of helimagnetic domain walls.It is now established that spin textures with non-trivial topology hold great technological potential enabling, e.g., reconfigurable logic gates, 11,25 race track memory, 9,10 and neuromorphic 26 / reservoir computing. 27Sensing of individual topological spin structures, and magnetic nano-objects in general, in a way that is compatible with the proposed device architectures and semiconductor fabrication methods, however, remains a challenging task.Topological spin arrangements have been resolved by various imaging techniques, including electron, 9,28 X-ray, 29,30 and magneto-optical 29 microscopy, as well as scanning tunneling microscopy, 30,31 magnetic force microscopy (MFM), 32,33 and nitrogen vacancy magnetometry. 34While these methods have provided important insight into the physics of topologically nontrivial spin textures, they are not directly transferrable to devices.For the specific case of skyrmion-electronics, a promising method is to utilize the topological Hall effect, 35,36 or magnetoresistance measurements, 37 but an expansion towards other magnetic nano-entities remains to be demonstrated.Thus, the development of dynamical and more agile read-out schemes that allow for resolving individual nanoscale spin textures in device-relevant geometries is highly desirable.
Here, we demonstrate how non-linear magnetic responses can be utilized to detect and identify both topologically trivial and non-trivial spin textures at the nanoscale.Combining MFM and micromagnetic simulations, we analyze the local magnetic response of magnetic disclinations, dislocations, and helimagnetic domain walls in the model system FeGe.Our results clarify the local magnetic properties and reveal characteristic fingerprints that enable selective detection of different nanoscale spin arrangements using superconducting micro-coils.The general feasibility and universality of this approach are demonstrated by two examples, considering the signal formation for an edge dislocation ( = − ) as well as a topologically trivial curvature domain wall ( = 0).

Local magnetic response at dislocations and domain walls
FeGe belongs to the family of helimagnets with B20 structure 2,3,[38][39][40] and its magnetic phase diagram is wellestablished. 41In addition, diverse nanoscale spin textures have been observed and their equilibrium structure has been analyzed in detail, rendering FeGe an ideal model system for this work.FeGe develops helimagnetic order below TN = 280 K, stabilized by the competition between Heisenberg exchange and the relativistic Dzyaloshinskii-Moriya interactions in the non-centrosymmetric crystal lattice.The helimagnetic ground state is characterized by a gradual rotation of the magnetization vector M about a wave vector  = ( 2  ) ̂, where  = 70 nm 42 is the helical period and  ̂ characterizes the direction of the helical axis (Fig. 1a).Fig. 1b shows an MFM image of FeGe in the helimagnetic phase, recorded in two-pass mode (see Experimental Section for details).Bright and dark lines reflect the periodic magnetic structure with λ  70 nm and q lying in the surface plane (white arrow) in agreement with neutron scattering data. 42Because of the lamellar morphology 18,19 -which is analogous to cholesteric liquid crystals -different types of defects arise in FeGe at the nanoscale, including topologically non-trivial objects such as dislocations (Fig. 1c) and zigzag disclination walls (Fig. 1e), respectively, as well as more simple curvature walls that do not carry a topological charge (Fig. 1d).Details about the inner structure of the different spin textures are reported elsewhere. 18,19Most importantly for this study, Fig. 1c-e reveal a universal feature that is shared by all defect structures, independent of their topology, shape and Figure 1.MFM imaging of helimagnetic order, dislocations and domain walls.a, Schematic illustration of the helical spin structure in FeGe described by the wave vector q and the characteristic stripe-like pattern probed by MFM in the helical phase.The colour scale indicates the direction of the magnetic stray field B from the sample.b, MFM image of the helimagnetic order within a single  domain in FeGe.Note that the bright and dark contrast originate from the spin helix, giving rise to a lamellar morphology with a measured periodicity of  70 nm, which is about three orders of magnitude smaller than the conventional stripe domains in ferromagnetic systems.In FeGe, domains are formed only on much larger length scales 19 as seen, e.g., in (d) and (e), corresponding to regions with a different orientation of .ce, MFM images showing magnetic dislocations in the lamellar-like spin structure (c), a curvature domain wall (d), and a zigzag domain wall composed of + and − disclinations (e).All 1D and 2D spin textures in (c) to (e) exhibit enhanced bright MFM contrast compared to the helimagnetic background.dimensionality: All defects exhibit an additionally enhanced contrast in the MFM data that is not observed in regions with perfect lamellar-like order (Fig. 1b), separating them from the helimagnetic background.A similar MFM response has been observed at the helimagnetic domain walls in earlier studies, but without clarifying the microscopic origin. 19Thorough examination of the data in Fig. 1 reveals that the enhanced MFM response is asymmetric: only the bright lines, which indicate an attractive force between probe tip and sample, exhibit increased intensity and width.Furthermore, we find that the enhanced MFM signal can be detected more than 100 nm above the surface, that is, before the actual spin structure of the defects is resolved (see Supplementary Fig. 1 and Supplementary Fig. 2).

Micromagnetic simulations
In order to understand the unusual local response of the magnetic defects, we conduct micromagnetic simulations.The MFM signal is proportional to the phase shift ∆ ∝ − 2   / 0 2 of the oscillating probe tip.Eint is the dipolar interaction energy between the tip and sample and  0 denotes the tip-sample distance.First, we theoretically discuss Eint on the level of linear response where the tip just acts as a probe and its influence on the magnetization of the sample is neglected: Here,   is the magnetization of the MFM tip,  is the magnetization in the sample, and χ d −1 is the dipolar interaction between the tip and the sample (see Supporting Note S1).  is often approximated by a point-dipole (  () =    3 ( −  0 );   is the magnetic moment of the tip and  0 its position).The stray field  of the helical phase is shown in Fig. 2a.It decreases exponentially as function of the distance 18 ,  ∝  −2 0 / , with /2 determining the decay length of the stray field, see Supplementary Note.Microscopically, B is generated by magnetic bulk charges, ∇ ⋅ , and surface charges,  ⋅  ( is the unit vector normal to the surface).As bulk charges are absent in the case of an ideal spin helix, B is dominated by surface charges, as shown in Fig. 2b.Thus, taking into account that the projected periodicity   of the lamella-like order increases whenever the helimagnetic structure is bent, i.e.   > , the generally stronger magnetic stray field B and phase shift ∆ at defects can be explained based on an increased decay length (~ −2 0 /  ) associated with local magnetic surface charges (see Supplementary Fig. 3, Supplementary Fig. 4, Supplementary Note).However, while this geometrically driven effect can give rise to higher attractive and repulsive forces at defect structures, it is qualitatively different from the asymmetric effect presented in Fig. 1, failing to explain why only attractive forces appear amplified at the defect sites.
This discrepancy leads us to the conclusion that the dipolar interaction between the magnetic moment of the tip and the stray field of the spin helix is nonnegligible. 24Therefore, we go beyond the linear response theory that describes non-invasive MFM measurements and include the local stray field of the magnetic MFM tip in our three-dimensional simulations, accounting also for emergent non-linear responses.We model the tip by a single dipole moment and the magnetization of FeGe is described by a lowest order gradient expansion: subject to the boundary conditions of the embedding helical phase in three spatial dimensions ( = /  is the normalized magnetization,   is the saturation magnetization, A is the exchange stiffness, D is the Dzyaloshinskii-Moriya interaction, and   is the dipolar stray field that is emitted by the tip).For simplicity, we neglect effects from the demagnetizing field and cubic anisotropies as they do not change the results qualitatively (see Methods for details on parameters and software).The calculated helimagnetic texture resulting under realistic experimental conditions is presented in Fig. 2c.As function of distance  0 , the stray field from the tip decays as   ∝  0 −3 in the dipole approximation; see Supplementary Note for a discussion of pyramidally shaped tips.As a consequence, the strongest polarizing effects in the spin helix are observed at the sample surface, where the tip induces an additional magnetic surface charge (blue in Fig. 2d).Vice versa, we find that the net induced surface charge is the main source for the magnetic stray field probed by the tip (bent black lines in Fig. 2c), leading to a stronger and more long-ranged attractive force (polynomially decaying instead of exponentially) compared to the unperturbed helimagnetic structure as displayed in Fig. 2a.Note that this net polarization of the magnetization occurs for helimagnetic order both with and without defect structures (see Supporting Note S1 for details).In summary, the magnetization of the tip  tip leads to an additional non-linear response in MFM.In a second order process,  tip creates magnetic charges within the sample via dipolar interactions which then feed back onto the tip: The efficiency for inducing magnetic charges, however, is set by the magnetic susceptibility χ, so that local variations in χ can lead to additional contributions in MFM.At the level of domains, such additional contributions have been studied intensively and are known as susceptibility contrast. 24Our calculations reveal that such susceptibility contributions are equally important at the nanoscale, leading to substantially different non-linear responses for spin-helix segments with different out-of-plane magnetization components.

Non-linear response at helimagnetic domain walls
To verify that the local susceptibility contrast observed at helimagnetic defects originates from tip-induced magnetic surface charges -i.e., a non-linear responsewe simulate MFM scans with oppositely magnetized tips.As an instructive example, we consider the case of a topologically non-trivial zigzag wall containing +π and -π disclinations. 19For reference, the magnetic surface charges of the ideal, undisturbed zigzag domain wall are presented in Fig. 3a.Simulations accounting for the tip-sample interaction are presented in Fig. 3b,d.
The simulations show that the enhanced magnetization of the +π and -π disclination centres inverts as the magnetization of the tip is switched from "down" (Fig. 3b) to "up" (Fig. 3d).In contrast, the lines connecting the +π and -π disclinations remain bright upon reversal, corresponding to an attractive force on the tip independent of the orientation of   ().
Corresponding experimental MFM scans on a real sample with "down" and "up" magnetized tip are presented in Fig. 3c and e, respectively (see Methods for experimental details).Both MFM images in Fig. 3c reveal qualitatively the same spin texture at the zigzag wall.In agreement with the simulations, we observe that the MFM signal at the disclination centre inverts (marked by white dashed circles).The signal associated with the domain wall, however, remains bright in both scans (marked by white dashed lines to the right of the circles).4][45] As the magnetization at defects deviates from the energetically favourable helical structure, it is reasonable to assume that it is more susceptible to external magnetic fields.The higher susceptibility leads to a more efficient generation of surface charges than in the helimagnetic background and, hence, an additional long-ranged attractive force, consistent with our MFM measurements (Supplementary Fig. 1, 2, 5-8).In contrast to the MFM contributions from magnetic surface charges, however, the local susceptibility contrast is proportional to   2 (Supplementary Note).
As a consequence, susceptibility-related signals linked to the zigzag wall do not invert along with the tip magnetization and can be isolated by adding MFM images gained with opposite tip magnetization.The latter is presented in Supplementary Fig. 9, 10 and 11 where the MFM sum image reveals a pronounced contrast associated with the curved helix structure, confirming that defects in the helimagnetic structure in FeGe exhibit a locally enhanced susceptibility.

Generalization of the detection scheme
In summary, we have identified a pronounced nonlinear response at defects of helimagnetic order in FeGe due to their specific surface polarization induced by the magnetic tip (Fig. 2).This non-linear response occurs as a magnetic field is applied, providing an additional opportunity for sensing and distinguishing nanoscale spin textures.In Figure 4, the latter is shown for selected examples of 1D-and 2D-defects; that is, a magnetic edge dislocation (topologically equivalent to a half-skyrmion or meron) and a topologically trivial curvature wall.
The basic detection scheme is presented in Fig. 4a, displaying an artistic view of a SQUID (superconducting quantum interference device) coil with a diameter of r = 500 nm comparable to state-ofthe-art SQUID-on-tip technology. 46,47uch nanoSQUIDs can be operated both without and with magnetic background fields, facilitating a sensitivity of 0.6 μB Hz −1/2 at 1 T as demonstrated in ref. 48Thus, using SQUIDs it is feasible to measure both linear and nonlinear responses, detectable as a variation in the magnetic flux ∆Φ.To quantify and compare the expected signals, we assume an idealized geometry where a magnetic field of 100 mT is produced by the same coil as used for detection (Fig. 4b-e).We note, however, that this field strength may be technologically challenging to realize with such SQUIDs.A conceptually equivalent alternative is to generate the local background field via patterned magnetic elements. 49The calculations show that for a magnetic dislocation, which binds a finite magnetic surface charge, both linear (0 mT) and non-linear (100 mT) detection is possible, yielding comparable changes in the magnetic flux (Fig. 4c).The 180 phase jump, which is induced as dislocations move through the helimagnetic background, 18 however, is more pronounced when applying non-linear detection.In contrast to edge dislocations, curvature domain walls exhibit alternating surface charges that macroscopically average to zero, so that the linear signal can effectively cancel out.However, these defects are then still detectable via the non-linear interaction (Fig. 4e) which generates a clear peak at the domain wall.This peak makes the signal asymmetric so that it remains detectable even when its oscillating fine-structure cannot be resolved.Based on the calculations, a signal span in magnetic flux of ∆Φ  10 -16 Wb ≈ 0.2 Φ0 is expected.For the coil in Fig. 4, this difference translates into a magnetic field ∆Φ / πr 2 ≈ 510 µT, which is readily measurable using SQUID magnetometers.Importantly, the variation in magnetic flux presented in Fig. 4c,e is specific to the cases depicted in Fig. 4b,d; in general, the measured signal depends on the coil geometry and position, as well as the direction of movement relative to the spin texture, providing additional information about the magnetic order at the nanoscale.For example, the magnetic flux is constant for translations perpendicular to the q-vector of the spin helix, but varies in the direction parallel to q, allowing to resolve phase jumps (Fig. 4c) and spatial modulations (Fig. 4e) in the helimagnetic spin structure.
On the one hand, the proposed detection scheme is compatible with local imaging techniques such as scanning SQUID microscopy, 50 where the coil is scanned across the sample surface to detect the defects, removing the requirement of sub-100 nm resolution to verify emergent defect structures.On the other hand, race-track like geometries 9,51 with stationary coils are possible, sensing moving edge dislocations, domain walls and other mobile topological defects via their nonlinear response.

Conclusions
The results presented in this work clarify the interaction of topologically non-trivial spin textures and domain walls in chiral magnets with external magnetic field, revealing a pronounced non-linear response due to field induced surfaces charges.The additional surface charges lead to a more long-ranged interaction compared to the unperturbed magnetic state, facilitating new opportunities for the detection and differentiation of nanoscale spin textures using nanoSQUIDs.Based on the non-linear response, the detection sensitivity can be improved as demonstrated for two instructive examples, i.e., a magnetic edge dislocation and a helimagnetic curvature wall.The proposed detection scheme is universal and, in principle, can be applied to all magnetic nano-objects that exhibit a different susceptibility than their surroundings, enabling sensing of otherwise hidden 1D and 2D spin textures.The latter is supported by the recent observation of enhanced magnetic susceptibility at antiferromagnetic domain walls in the topological insulator MnBi2Te4, 45 expanding application opportunities into the realm of antiferromagnetic spintronics.Thus, in addition to the fundamental insight into the nanoscale physics of chiral magnets, this work introduces a viable read-out scheme for topological magnetic defects, and local spins arrangements in general, opening new possibilities for the design and fabrication of on-chip devices for spintronics.

Methods
Sample Preparation: Single crystals of FeGe were grown by the chemical vapor transport method.To achieve flat high-quality surfaces for MFM imaging, the samples were prepared by lapping and polishing to achieve a root mean square roughness of approximately 1 nm and cleaned with high-purity acetone and methanol, following the same procedure as described in Refs. 18,19.

Magnetic force microscopy:
The MFM data was recorded using a commercial SPM system (NT-MDT NTEGRA Prima AFM).Magnetic probe tips (PPP-MFMR from Nanosensors) with force constant of 2.8 N/m and quality factor Q of about 200 were used.The tips possess a hard magnetic coating with effective magnetic moment of 10 -16 A m 2 and have been magnetised by a permanent magnet prior to the measurements.Sample cooling was achieved using a water-cooled Peltier element and the measurements were carried out in N2 atmosphere (with a few mBar overpressure) to prevent ice formation.The microscope was operated in two-pass MFM mode with the magnetic tip oscillating at its resonance frequency ( 70 kHz) with an amplitude of ≈ 30 nm.During the first pass, a topography image was collected by scanning with the tip close to the sample surface.During the second pass, the tip was lifted 10-200 nm (in addition to 30 nm in the first pass) and retraced the measured topography to sense solely the magnetic interaction between the magnetic stray field of the sample and the magnetic tip.Between measurements with opposite magnetisation of the tip, the sample was heated to room temperature and the protective hood was removed to switch the tip magnetisation using a permanent magnet.Regions of interest were tracked using an optical microscope and the obtained MFM images were aligned using topographical features on the sample surface.

Simulations:
We model the magnetization in FeGe in the presence of an MFM tip in an effective isotropic model.We assume that the magnetization of the MFM tip and hence also the stray field are constant.This model is similar to the micromagnetic model, but neglects contributions of the demagnetizing field within the sample as these significantly slow down our calculations while their corrections are assumed to be only marginal.The simulations were performed at T = 0 K using the following parameters for FeGe:  = 8.78 pJ m −1 ,  = 1.58 mJ m −2 , and   = 384 kA m −1 . 52For the numerics, we discretize the continuum theory on a regular mesh of cuboids (  ,   ,   ) with   ≈ /16 where  = 4/ is the wavelength of the helix.The rather coarse discretization is still suitable as we approximate derivatives by fourth order stencils. 53The energy of the 3d setup is minimized by a (single precision) GPU-accelerated self-written software 20 where one boundary condition is von-Neumann (the surface to vacuum) and the opposite boundary is fixed to the bulk minimizer.The other boundaries are also fixed but were pre-relaxed under the constraints of one "bulk" and one "surface" edge.The thickness of the 3d slab is usually of the order   ≈ 2 with additional tests for   ≈ 8.For one pixel in a non-linear MFMimage, we relax the magnetization in a local spheroid for three different tip heights which we then use to compute the second derivative of the dipolar interaction energy.

Figure 2 .
Figure 2. Calculated local response of the helimagnetic spin structure.a, Side view of the helimagnetic order which -in the absence of an invasive magnetic tip -differs only slightly for the surface (top) and the bulk (bottom).The out-of-plane magnetic components associated with the helical spin structure (sketched by solid black arrows) generate alternating magnetic surface charges (blue to red) with a periodicity of about 70 nm, which are the main source for the magnetic stray field (curved black arrows, saturation encodes field strength).b, same as in (a) in the presence of an invasive magnetic tip (  = 10 −16 A m 2 , positioned at distance  0 = 40 nm from the surface), leading to substantial changes in the helimagnetic structure at the surface (top) compared to the bulk (bottom).The extra field of the tip in (b), approximated by a single dipole, is coloured orange.c, top view of the alternating magnetic surface charges seen in (a).d, same as in (c) in the presence of an invasive magnetic tip.The position of the MFM tip is indicated in (d) by white rings, each corresponding to a factor 2 decreased magnetic field.The polarizing influence of the tip in (b), (d) is clearly visible.

Figure 3 .
Figure 3. Magnetic response from a zigzag domain wall with alternating + and − disclinations as function of the orientation of the tip magnetization.a, Calculated magnetic surface charges of a zigzag domain wall at the surface of FeGe.The colour denotes the outof-plane magnetization related to the spin helix from pointing up (blue) to down (red).b, d, Calculated non-linear MFM response for a down (red, b) and up (blue, d) magnetized tip, taking the tip-sample interaction into account (lift height: 100 nm, tip moment: 2 ⋅ 10 −16 A m 2 ).Bright and dark colours indicate attractive and repulse forces, respectively.The pattern of bright and dark lines associated with the spin helix inverts as the tip changes magnetization direction, whereas an additional attractive force is detected at the domain wall position independent of the tip magnetization.c, e, Corresponding MFM images of a zigzag domain wall recorded at the same position with (c) tip magnetized down and (e) up.The size of the scanning area is 1 μm × 1 μm.The white dashed circles mark the centre of the disclination, and the white dashed lines mark the domain wall.

Figure 4 .
Figure 4. SQUID-based read-out scheme for the detection of 1D and 2D magnetic spin textures.a, Illustration of a magnetic track and schematic SQUID coil (diameter: 500 nm) positioned 10 nm above the surface, presenting the basic setup for detection.Using stationary coils, mobile topological spin textures -here, a dislocation -can be sensed and counted via a defect-specific change of the magnetic flux through the coil.b, Out-of-plane magnetization, mz, of an edge dislocation under the influence of the magnetic stray field (100 mT) from the coil illustrated by the yellow dashed line.B gives the direction of the magnetic field within the coil and the black arrow indicates the direction of motion relative to the edge dislocation.c, Induced magnetic flux measured with biased (100 mT) and non-biased (0 mT) coils.d, and e, Same as in (b) and (c) for a curvature wall.